The 7 Most Important Equations for Your Retirement E-Book

Moshe A. Milevsky

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Sprache: Englisch

Seitenzahl: 269

Veröffentlichungsjahr: 2012

Beschreibung

The 800 years of scientific breakthroughs that will help salvage your retirement plans Physics, Chemistry, Astronomy, Biology; every field has its intellectual giants who made breakthrough discoveries that changed the course of history. What about the topic of retirement planning? Is it a science? Or is retirement income planning just a collection of rules-of-thumb, financial products and sales pitches? In The 7 Most Important Equations for Your Retirement...And the Stories Behind Them Moshe Milevsky argues that twenty first century retirement income planning is indeed a science and has its foundations in the work of great sages who made conceptual and controversial breakthroughs over the last eight centuries. In the book Milevsky highlights the work of seven scholars--summarized by seven equations--who shaped all modern retirement calculations. He tells the stories of Leonardo Fibonnaci the Italian businessman; Benjamin Gompertz the gentleman actuary; Edmund Halley the astronomer; Irving Fisher the stock jock; Paul Samuelson the economic guru; Solomon Heubner the insurance and marketing visionary, and Andrey Kolmogorov the Russian mathematical genius--all giants in their respective fields who collectively laid the foundations for modern retirement income planning. * With baby boomers starting to hit retirement age, planning for retirement income has become a hot topic across the country * Author Moshe Milevsky is an internationally-respected financial expert with the knowledge you need to assess whether you are ready to retire or not * Presents an entertaining, informative narrative approach to financial planning Understanding the ideas behind these seven foundation equations--which Moshe Milevsky explains in a manner that everyone can appreciate--will help baby boomers better prepare for retirement. This is a book unlike anything you have ever read on retirement planning. Think Suze Orman meets Stephen Hawking. If you ever wondered what the point of all that high school mathematics was, Moshe Milevsky's answer is: So that you can figure out how to retire...while you can still enjoy your money.

Leseprobe

Table of Contents

Other books by Moshe A. Milevsky

Title Page

Copyright

Dedication

Introduction: An Equation Can’t Predict Your Future … But It Can Help You Plan for It

Reality Check

Chapter 1: How Long Will My Number Last?

Equation #1: Leonardo Fibonacci (1170–1250)

The Spending Rate: A Burning Question

Fibonacci’s Fabulous Flash of Finance

Manipulating the First Equation

Is It Really His?

Can We Really Know Interest Rates?

Back to Fibonacci’s Life Story

Okay, Here Come the Rabbits

He Retired Wealthy

Chapter 2: How Long Will I Spend in Retirement?

Equation #2: Benjamin Gompertz (1779–1865)

Simplistic Retirement Planning

Gompertz’s Big Discovery

So How Long Should I Plan for in Retirement?

Back to Benjamin Gompertz

Chapter 3: Is a Pension Annuity Worth It?

Equation #3: Edmond Halley (1656–1742)

Take the Pension or a Lump Sum?

Halley’s History, Part I

Pension Value: Age and Interest

Back to Halley’s History, Part II

The Great Comet

Chapter 4: What is a Proper Spending Rate?

Equation #4: Irving Fisher (1867–1947)

Retirement Inflation: It’s Getting Personal

Sustainable Spending Rates: Maybe

The Fourth Equation, Explained

Fisher’s Optimal Retirement Plan, Applied

In Fisher’s Words

Confirming the Numbers

Irving Fisher’s Rise, Fall and Rise Again

The 1929 Crash

Chapter 5: How Much in Risky Stocks versus Safe Cash?

Equation #5: Paul Samuelson (1915–2009)

Time Is on Your Side?

Human Capital: You Are Wealthier than You Think

The Equation, Explained

The Equation, Applied: Case Studies and Examples

Retirees Should Protect Their Equities

Paul Samuelson’s Life and Impact

Chapter 6: What Is Your Financial Legacy Today?

Equation #6: Solomon S. Huebner (1882–1964)

Human Life Value: Raison d’être of Life Insurance

What’s the Most Effective Way to Create a Legacy?

Life Insurance: The Formula

More Examples: Different Ages

If You Try This at Home

He Wasn’t a Quant

Secondary Markets and Life Settlements

Term versus Whole Life

But Sunny Sol Was a Fan of Annuities Too

The Historical Huebner versus Fisher

Chapter 7: Is My Current Plan Sustainable?

Equation #7: Andrei N. Kolmogorov (1903–1987)

Will Your Retirement Plan Work Out?

A Purely Imaginary Case to Help

From Monte Carlo to Netherlands, Sweden and then Russia

The Equation Itself Is Only Partial

Detailed Example

Verification that Kolmogorov’s Equation Is Satisfied

Physics, Not Finance

Back to Andrei Nikolaevich Kolmogorov

Conclusion: Controversies, Omissions and Concluding Thoughts

What Equations Are Missing?

Appendix: Crash Course on Natural and Unnatural Logarithms

References and Sources

Acknowledgments

About the Author

Short Poem by Maya Milevsky (age 11)

Index

Other books by Moshe A. Milevsky

Strategic Financial Planning over the Lifecycle: A Conceptual Approach to Personal Risk Management with N. Charupat and H. Huang (Cambridge, 2012)

Your Money Milestones (FT Press, 2010)

Pensionize™ Your Nest Egg with A. Macqueen (Wiley, 2010)

Are You a Stock or a Bond? (FT Press, 2009)

Lifetime Financial Advice: Human Capital, Asset Allocation and Insurance with R. Ibbotson, P. Chen and K. Zhu (CFA Institute, 2007)

The Calculus of Retirement Income (Cambridge, 2006)

Wealth Logic (Captus, 2002)

Insurance Logic with A. Gottesman (Stoddart, 2002)

Money Logic with M. Posner (Stoddart, 1999)

Copyright © 2012 by Moshe A. Milevsky

All rights reserved. No part of this work covered by the copyright herein may be reproduced or used in any form or by any means—graphic, electronic or mechanical—without the prior written permission of the publisher. Any request for photocopying, recording, taping or information storage and retrieval systems of any part of this book shall be directed in writing to The Canadian Copyright Licensing Agency (Access Copyright). For an Access Copyright license, visit www.accesscopyright.ca or call toll free 1-800-893-5777.

Care has been taken to trace ownership of copyright material contained in this book. The publisher will gladly receive any information that will enable them to rectify any reference or credit line in subsequent editions.

The material in this publication is provided for information purposes only. Laws, regulations, and procedures are constantly changing, and the examples given are intended to be general guidelines only. This book is sold with the understanding that neither the author nor the publisher is engaged in rendering professional advice. It is strongly recommended that legal, accounting, tax, financial, insurance, and other advice or assistance be obtained before acting on any information contained in this book. If such advice or other assistance is required, the personal services of a competent professional should be sought.

Library and Archives Canada Cataloguing in Publication Data

Milevsky, Moshe Arye, 1967-

The 7 most important equations for your retirement : the fascinating people and ideas behind planning your retirement income / Moshe A. Milevsky.

Includes bibliographical references and index.

ISBN 978-1-1182915-3-5

1. Retirement—Planning. 2. Financial planners—Anecdotes.

I. Title. II. Title: Seven most important equations for your retirement.

HG179.M51846 2012332.024’014C2012-901735-3

ISBN 978-1-11834740-9 (ebk); 978-1-11834741-6 (ebk); 978-1-11834742-3 (ebk)

Production Credits

Cover design: Ian Koo

John Wiley & Sons Canada, Ltd.

6045 Freemont Blvd.

Mississauga, Ontario

L5R 4J3

www.wiley.com

To the memory of my grandfather, Rav Hillel Mannes, PhDc; a German gentleman, scholar and orthodox Jew.

Introduction

An Equation Can’t Predict Your Future … But It Can Help You Plan for It

Most books about retirement planning are written as guides, instruction manuals or “how-to” books. The authors tell you what to do, when to do it, and what to expect. I know this quite well because I have authored many such tomes myself.

Rest assured, this is not one of those books.

You see, time is running out. North American baby boomers are getting within shouting distance of their golden years. Most have finally grasped that—despite the dreamy commercials and brochures—retirement isn’t a long vacation that begins at the mythical age of 65. It’s a gradual winding-down process, possibly involuntary, with austere financial implications. The timeworn questions about proper savings rates, the best mutual funds or the ideal size of your nest egg number have been pushed aside by a pragmatic economic reality: “This is what I have, give or take a few more years of saving. How do I make it last?”

In other words, it is time to have some conversations about retirement income planning, also known as de-accumulation planning. The stories in this book should lead you into those conversations. These conversations should take place with your family and loved ones and possibly with a professional financial advisor as well (whether you love them or not).

Reality Check

Today, there are two cold, hard facts about aging consumers heading into the second decade of the 21st century: i) most are not saving enough money to maintain their current standard of living, and ii) many are financially illiterate. Alas, study after published study continues to document a shocking lack of knowledge about basic financial affairs and a correspondingly miniscule financial cushion for retirement.

These two problems are obviously linked.

Yes, I know from many years of teaching experience that financial conversations are often dry and humorless. So I promise to do my best to lighten up the topic by keeping the technicalities to a minimum and focusing on the art.

“Art,” you say?

Yes. In my mind, famous equations are like beautiful Picassos. Even if I don’t quite understand the painting or the mathematics I can certainly appreciate the beauty and genius behind it. The seven equations presented in this book typify, at least for me, the conciseness, elegance and beauty that the best of the best equations demonstrate. By the end of this book, if you’re not already inclined to appreciate mathematical equations for what they are, I hope you’ll agree about the beauty.

Here’s a brief outline of what’s ahead.

In the first chapter, I will start by asking a simple question: How long will your nest egg last, if you were to stop contributing today, and instead withdraw a fixed amount each year while earning a fixed interest rate each year for the rest of your life? Although neither of these fixes are realistic in practice, this sort of analysis provides a quick and sobering assessment of whether you can maintain your standard of living, or when the money will run out if you can’t. The underlying mathematics of this first equation is rather simple, and does not require any complex algorithms. Along the way, you’ll learn about the 13th-century Italian educator and mathematician Leonardo Fibonacci and his remarkable technique for solving such financial problems.

In the second chapter, I’ll address the length of life itself. Here is the motivating question: Given your family history, current lifestyle and recent medical experience, what are the chances that you, your spouse, or both of you, will reach age 90, 95 or even 100? In this chapter you’ll learn about longevity risk and learn about the work of famous British actuary Benjamin Gompertz. He discovered and formulated the first natural law of human mortality almost 200 years ago. His equation is used daily by medical researchers, demographers and insurance specialists.

The third chapter focuses on life annuities and pensions. This is the opportunity to inquire: Do you have a true pension? I’ll encourage you to dig deep into the mechanics of your employer or government retirement plan. Many are called pensions, but are basically pooled savings plans. You can however use your retirement money to buy a private pension annuity, but it can be surprisingly expensive and there are economic tradeoffs to consider. You’ll also learn about the famous 17th-century astronomer and Savilian Professor of Geometry at Oxford University, Edmond Halley, who developed the first mathematical expression for the price of a life annuity. Although his full-time job was mapping the solar system and searching for comets, it seems he also had a soft spot for pensions and annuities.

In Chapter Four, I’ll invite you to ponder patience. How important is it for you to enjoy your retirement money earlier rather than later? The financial industry talks about replacing 70% to 80% of your income in retirement and spending no more than 4% of your nest egg each year of retirement. But where does that number come from? Do you really want the exact same standard of living, regardless of how long you live? Or, are you willing to scale back if in fact you live to a ripe old age? This is yet another tradeoff, and related to the concept called subjective time preference. In this chapter, I argue one size doesn’t fit all. I’ll introduce you to the celebrated 1920s American economist and pulpit professor Irving Fisher. He formulated the precursor to financial lifecycle models and cautioned against the ravages of inflation, yet another important concern for retirees.

In Chapter Five, I address financial risk head-on. How comfortable are you with the stock market? The last quarter century has taught us that time doesn’t necessarily diversify away risk. Stocks—which fluctuated more than bonds—earned less than bonds since the late 1980s. Whether this trend will persist going forward is debatable. But if stocks do continue their lackluster performance—and the economy sputters—will this impact the remaining working years of your job? In the language of financial economics, are your human capital and financial capital intertwined? If so, you might want to lay off the stocks as you get closer to retirement. Asset allocation should involve your entire personal balance sheet. In this chapter you will learn about Professor Paul Samuelson, the first American to win the Nobel Prize in Economics in 1970. He introduced an asset allocation equation that’s relevant over the entire lifecycle, but especially during retirement. He was also a fierce opponent of something called time diversification and anyone who dared support it.

In Chapter Six, I focus your attention beyond retirement to the next generation. What do you plan to leave the kids? Is legacy a priority for you? Many who are close to or in retirement often dedicate a particular account, asset or sum for their loved ones. It’s not uncommon to hear that the house is going to the kids, or that the investments will eventually go to the grandkids. This is commendable, but is it financially affordable? Here is yet another tradeoff. In fact, if legacy is your motive then you might want to see what life insurance can do for you. And vice versa: if you already have life insurance (paid up, or not) but need the money earlier, you might want to think about giving it up. Legacy and life insurance are important conversations to have, and this chapter is the opportunity to have those discussions. The founding father of the economics of life insurance was the charismatic teacher Dr. Solomon Huebner (“Sunny Sol” to his students) and this chapter will tell his story. You will learn about the most important equation in the life insurance business, which is naturally quite important for retirement income planning as well.

The final chapter, Chapter Seven, will pull all these strands together. I will present the one final unified equation that measures the sustainability of your retirement plan. It takes into account your age, current asset allocation, pension income, longevity and everything else on your personal balance sheet. It’s a one-number summary, and the person responsible for this final equation is a Russian mathematical prodigy named Andrei Nikolaevich Kolmogorov. His life story takes us up the Volga and into the venomous politics of Stalin and the Russian Academy of Sciences.

Although the formal concept of retirement is a relatively modern phenomenon—dating to German Chancellor Otto von Bismarck’s creation of the state pension, and American President Franklin D. Roosevelt’s Social Security—the mathematical and statistical science behind retirement income planning is centuries old.

Looking into the future, I don’t know if governments, corporations and individuals can afford to continue paying for society’s current retirement promises. But one thing is certain: these seven equations and the science behind them will play a central role in the years beyond three score and 10, for many centuries to come. I hope you enjoy the mathematical, financial and historical expedition that lies ahead. I know I did.

Chapter 1

How Long Will My Number Last?

Equation #1: Leonardo Fibonacci (1170–1250)

Leonardo had a problem. A close friend had invested some money a few years earlier in a local Italian bank, in Pisa, that promised him steady interest of 4% per month. (Yes, I wish I got 4% interest per month. I don’t even get that per year nowadays. Sounds shady to me.) Anyway, rather than sitting by and letting the money rapidly grow and compound over time, Lenny’s friend started withdrawing large and irregular sums of money from the account every few months. These sums were soon exceeding the interest he was earning and the whole process was eating heavily into his capital. To make a long story short, Leonardo—known to be quite good with numbers—was approached by this friend and asked how long the money would last if he kept up these withdrawals. Reasonable question, no?

Now, if Leonardo had been me, he’d have pulled out his handy Hewlett-Packard (HP) business calculator, entered the cash flows, pushed the relevant buttons and quickly had the answer. In fact, with any calculator these sorts of questions can be answered quite easily using the technique known as present value analysis—something all finance professors teach their students on the first day of class. Later, I’ll explain this important process in some detail.

Unfortunately, Leonardo didn’t have access to an HP business calculator that performed the necessary compound interest calculations. (He didn’t have a calculator at all because they hadn’t been invented yet.) You see, Leonardo was asked this question more than 800 years ago, in the early part of the 13th century. But to answer the question—which he certainly did—he actually invented a technique we know today as present value analysis. Yes, the one I mentioned we teach our students.

You might have heard of Leonardo by his more formal name: Leonardo Pisano filius (“family” in Latin) Bonacci, a.k.a. Fibonacci to the rest of the world, and probably the most famous mathematician of the Middle Ages.

In fact, Fibonacci helped solve his friend’s problem—writing the first commercial mathematics textbook in recorded history in the process—and introduced a revolutionary methodology for solving complicated questions involving interest rates. Let me repeat: his technique, with only slight refinements, is still used and taught to college and university students 800 years later. Now that is academic immortality! (He published and his name hasn’t perished yet.)

Everyone owes a debt of gratitude to Fibonacci. Had it not been for him, we would probably still be using Roman numerals in our day-to-day calculations. He helped introduce and popularize the usage of the Hindu–Arabic number system—the 10 digits from zero to nine—in the Western world by illustrating how much easier they were for doing commercial mathematics. Imagine calculating square roots or performing long division with Roman numerals. (Okay: What is XMLXVI times XVI?) Well, you can thank Leonardo.

Leonardo Fibonacci was the first financial engineer, or “quant” (translation: highly compensated, scary-smart guys and gals who use advanced mathematics to analyze financial markets) and he didn’t work on Wall Street or Bay Street. He worked in the city of Pisa. More on his well-known work, and lesser-known life, later.

The Spending Rate: A Burning Question

Let’s translate Fibonacci’s mostly hypothetical 800-year-old puzzles into a problem with more recent implications. Imagine you’re thinking about retiring and have managed to save $300,000 in your retirement account. For now, I’ll stay away from discussing taxes and the exact administrative classification of the account. (I’ll revisit this case in Chapter Seven, where I’ll add more realistic details.) Allow me to further assume you’re entitled to a retirement pension income of $25,000 per year. This is the sum total of your (government) Social Security plus other (corporate) pension plans—but the $25,000 is not enough. You need at least $55,000 per year to maintain your current standard of living. This leaves a gap of $30,000 per year, which you hope to fill with your $300,000 nest egg. The pertinent question, then, becomes: Is the $300,000 enough to fill the budget deficit of $30,000 per year? If not, how long will the money last?

As you probably suspected, your $300,000 nest egg is likely not enough. Think about it this way: the ratio of $30,000 per year (the income you want to generate) divided by the original $300,000 (your nest egg) is 10%. There is no financial instrument I’m aware of—and I’ve spent the last 20 years of my life searching for one—that can generate a consistent, guaranteed and reliable 10% per year. If you don’t want to risk any of your hard-earned nest egg in today’s volatile economic environment, the best you can hope for is about 3% after inflation is accounted for, and even that is pushing it. Sure, you might think you’re earning 5% guaranteed by a bank, or 5% in dividends or 5% in bond coupons, but an inflation rate of 2% will erode the true return to a mere 3%. Needless to say, 3% will only generate $9,000 per year in interest from your $300,000 nest egg. That is a far cry from (actually $21,000 short of) the extra $30,000 you wanted to extract from the nest egg.

You have no choice. In retirement you will have to eat into your principal.

Here’s a side note. In my personal experience talking to retirees and soon-to-be retirees, I find this realization is one of the most difficult concepts they must accept. Some people simply refuse to spend principal and instead submit to a reduced standard of living. Principal is sacred and they agree to live on and adapt to interest income. But in today’s low-interest-rate environment, once you account for income taxes, living on interest only will eventually lead to a greatly reduced standard of living over time.

Once you accept that actually depleting your nest egg is necessary, the next—and much more relevant—question becomes: If I start depleting capital, how long before there is nothing left? After all, if you eat into the $300,000 there’s a chance it might be gone, especially if you live a long time. This is exactly where Leonardo Fibonacci’s insight and technique come in handy, and why I’ve bequeathed to him Equation #1.

Time to roll up the sleeves and get to work. Let’s plug some numbers into Equation #1 and see what Fibonacci has to say.

(You might want to quickly flip back to the equation at the start of the chapter.)

Notice the right-hand side lists three variables (or inputs) that can affect the outcome. The first is the letter W, which represents the size of your nest egg, $300,000. The second variable is c, which captures the amount you would like to spend or consume, above and beyond any retirement pension you might be receiving. (This was $30,000 in the earlier example.) Although your spending takes place continuously (daily, weekly) it adds up to the value of c, per year. Think of it as a rate. The final variable, r, the trickiest to estimate, is the interest rate your nest egg is earning while it’s being depleted, expressed in inflation-adjusted terms. That was the 3% number I mentioned earlier. Now all that’s left is to compute the natural logarithm of the ratio, denoted by ln[] in the first equation.

Natural logarithms are close cousins of common logarithms. Both buttons appear on any good business calculator, but the latter uses a base of 10 and the former a base of 2.7183. If you’re unfamiliar with natural logarithms—or it has been a while since high-school mathematics—you can find a crash course on natural logs and how they differ from common logs in the appendix to this book.

For now, you can think (very crudely) of the natural logarithm as a process that shrinks numbers down to a compact size that is much easier to work with. Later you can worry about how exactly this shrinking works.

In words, here is the harsh truth. Keep up this lifestyle, and you’ll be broke by the beginning of the 12th year of retirement spending. Not a good outcome, although you will still get your $25,000 pension for the rest of your life, which may (or may not) be enough. But the nest egg is blown. Don’t feign surprise. You knew that a yearly $30,000 withdrawal (i.e., spending from the nest egg) would be too much if all you’re earning is 3%. But what if you lower the withdrawal rate? Again, Fibonacci’s equation, Equation #1, divulges exactly how many years of income you’ll gain if you cut down on your planned spending.

The input choices are infinite (no pun intended), so to help you get a better sense of the resulting values I’ve attached two tables with a range of output numbers. Table 1.1 assumes an investment return of 1.5%, adjusted for inflation, while Table 1.2 assumes a higher (3%) investment return, also adjusted for inflation. Again, you might think these are rather small numbers but remember these numbers are net of inflation, or what I call “real rates.” If your bank is paying you 3% on your savings account, but inflation erodes 2% per year, then all you’re really earning is (approximately) 1%.

Table 1.1 In How Many Years Will the Money Run Out If You Are Earning 1.5% Interest?

Some might argue this “equation”—nominal interest earned, minus inflation rate, equals real interest—is more important than all seven equations mentioned in this entire book! If you’re wondering, the person responsible for this insight is Irving Fisher, the early 20th-century American economist and champion of Equation #4. No rush. We’ll get to his story.

Back to the tables. The columns represent the size of your nest egg (W), and the rows represent the annual spending rate (consumption above any pension income)—also adjusted for inflation. Think of them as today’s dollars.

Looking at Table 1.1, if you start retirement with $300,000 in a bank account earning 1.5% interest every year and you plan to withdraw $35,000 every year, then according to Equation #1 the money will run out in exactly 9.2 years. This is 110 months of income. That’s it!

In contrast, if you reduce your spending withdrawals to $20,000 and start with the same $300,000 nest egg, your money will last 17 years. Sounds like a lot of time, but note if you retire at 65 this strategy will last (only) until you’re 82.

As you’ll see later in Chapter Two, when we explore patterns of longevity and mortality in retirement, there are better-than-even odds you’ll still be alive at age 82. So even $20,000 from a nest egg of $300,000 (a 6.66% initial spending rate) is too high, unless you’re willing to (only) live on the pension income of $25,000 once the money runs out of the nest egg. You might be willing to take that chance and trade off more money earlier in retirement in exchange for a reduced standard of living later in retirement—when you’re less likely to be alive—but again, that’s your choice to make after you know the numbers and odds. We will return to this economic tradeoff in Irving Fisher’s Chapter Four.

Now, let’s say you have $1 million in your bank account (earning 1.5%) and you plan to withdraw $50,000 per year. How long will the money last? Those values aren’t directly in the table. What do you do? Well, in this case you should be able to use the equation directly (which is actually the point of this book).

Alternatively, you’ll notice this equation scales in W and c. In other words, you can divide both W and c numbers by any number and the results don’t change. So whether you have $1 million and are withdrawing $50,000, or you have $500,000 and are withdrawing $25,000, or you have $2 million and are withdrawing $100,000, they’re the equivalent mathematical problem. (Although, personally, I’d obviously like to have the $2 million.) In all cases, the ratio of withdrawal-to-wealth is 1/20. Look carefully at Equation #1: only the ratio matters. On a side note, mathematicians love equations that scale. It helps do something called “reduce the dimensionality” of a problem, and eliminates the need for unnecessary information, so they tend to get excited about these things. Yes, geeky, I know.

Either way, Table 1.1 tells us that in this particular case, the money will last 23.8 years, assuming an interest rate of 1.5%.

Table 1.2 In How Many Years will the Money Run Out If You Are Earning 3% Interest?

As I encouraged earlier—and like all the other equations displayed in this book—you are now free to plug in your own withdrawal assumptions and interest rates.

The one thing you might wonder about is the odd-looking symbols in the upper right-hand corner of Table 1.2. They’re not stray symbols or typos, but actually represent the mathematical symbol for infinity. Don’t be scared. That is good news. Under these conditions the nest egg money will never run out.

Here’s an explanation for why the answer is defined to be infinity, in some cases. Skip ahead if you want. Look carefully at the three cells in which the infinity symbols appear and their corresponding row and column coordinates. In particular, when your nest egg is $500,000 the 3% interest rate will generate $15,000 in annual interest. This exceeds the $10,000 you would like to extract every year. So instead of the nest egg shrinking over time, it will continue to grow! Ergo, the money will never run out. In fact, if you withdraw or consume $15,000 from the account per year, exactly the interest you are earning, the account will continue at the same $500,000 value forever. The same concept applies to the $400,000 case, where the 3% interest will generate $12,000—more than the $10,000. In all these cases the denominator within the logarithm will either be zero or negative. The logarithm of infinity (if you divide something by zero) is infinity, and the logarithm of a negative number is simply undefined. So before you use the formula on your calculator, make sure you are spending (withdrawing) more money than the interest you are earning. Otherwise, Fibonacci’s equation might lead to gibberish.

Here’s the bottom line with infinites. May we all be lucky enough to have large enough nest eggs relative to our withdrawal rate that Fibonacci’s equation results in infinity. Most of us, unfortunately, will retire to a reality reflected in the lower left-hand corner of these tables.

Fibonacci’s Fabulous Flash of Finance

The name Fibonacci is widely recognized among the bookish masses for something known as the Fibonacci series (or Fibonacci numbers), which has nothing to do with retirement finance or stock trading and more to do with sexually active rabbits. More on this later, but first let me describe Fibonacci’s contributions to commercial mathematics.

To begin with, Leonardo Pisano—a.k.a. Fibonacci—wrote a very famous book called Liber Abaci